Semisoft elastic response of nematic elastomers to complex deformations

Active Materials for Functional Origami

In recent decades, origami has been explored to aid in the design of engineering structures. These structures span multiple scales and have been demonstrated to be used toward various areas such as aerospace, metamaterial, biomedical, robotics, and architectural applications. Conventionally, origami or deployable structures have been actuated by hands, motors, or pneumatic actuators, which can result in heavy or bulky structures. On the other hand, active materials, which reconfigure in response to external stimulus, eliminate the need for external mechanical loads and bulky actuation systems. Thus, in recent years, active materials incorporated with deployable structures have shown promise for remote actuation of light weight, programmable origami. In this review, active materials such as shape memory polymers (SMPs) and alloys (SMAs), hydrogels, liquid crystal elastomers (LCEs), magnetic soft materials (MSMs), and covalent adaptable network (CAN) polymers, their actuation mechanisms, as well as how they have been utilized for active origami and where these structures are applicable is discussed. Additionally, the state-of-the-art fabrication methods to construct active origami are highlighted. The existing structural modeling strategies for origami, the constitutive models used to describe active materials, and the largest challenges and future directions for active origami research are summarized.

Surface Instability in a Nematic Elastomer

Liquid crystal elastomers (LCEs) are soft phase-changing solids that exhibit large reversible contractions upon heating, Goldstone-like soft modes, and resultant microstructural instabilities. We heat a planar LCE slab to isotropic, clamp the lower surface, then cool back to nematic. Clamping prevents macroscopic elongation, producing compression and microstructure. We see that the free surface destabilizes, adopting topography with amplitude and wavelength similar to thickness. To understand the instability, we numerically compute the microstructural relaxation of a "nonideal" LCE energy. Linear stability reveals a Biot-like scale-free instability, but with oblique wave vector. However, simulation and experiment show that, unlike classic elastic creasing, instability culminates in a crosshatch without cusps or hysteresis, and is constructed entirely from low-stress soft modes.

Rate-dependent stress-order coupling in main-chain liquid crystal elastomers

Liquid crystal elastomers (LCEs) exhibit significant viscoelasticity. Although the rate-dependent stress-strain relation of LCEs has already been widely observed, the effect of the intricate interplay of director rotation and network extension on the viscoelastic behavior of main-chain LCEs remains inadequately understood. In this study, we report real-time measurements of the stress, director rotation, and all strain components in main-chain nematic LCEs subjected to uniaxial tension both parallel and tilted to the initial directors at different loading rates and relaxation tests. We find that both network extension and director rotation play roles in viscoelasticity, and the characteristic relaxation time of the network extension is much larger than that of the director rotation. Interestingly, the gradual change of the director in a long-time relaxation indicates the director reorientation delay is not solely due to the viscous rotation of liquid crystals but also arises from its coupling with the highly viscous network. Additionally, significant rate-dependent shear strain occurs in LCEs under uniaxial tension, showing non-monotonic changes when the angle between the stretching and the initial director is large enough. Finally, a viscoelastic constitutive model, only considering the viscosity of the network by introducing multiplicative decomposition of the deformation gradient, is utilized to manifest the relation between rate-dependent macroscopic deformation and microscopic director rotation in LCEs.

A Programmable Liquid Crystal Elastomer Metamaterials With Soft Elasticity

Liquid crystal elastomers (LCEs) are a rubbery network of polymers with ordered liquid crystal mesogens. The combination of rubber elasticity and the anisotropic liquid crystalline order gives exceptional mechanical properties, like soft elasticity, where near-constant stress accompanies large elastic deformation in the material. However, the soft elasticity in LCEs is often bounded by the intrinsic molecular interactions and structures, limiting the range of programmable mechanical properties and functionalities. Here, we demonstrate that the semi-soft elasticity of LCEs can be integrated into the framework of metamaterials to realize markedly programmabilities. Under uniaxial deformation, each state of the building blocks in metamaterials and the molecular composition of the nematic LCEs is associated with a distinctly different stress-strain relation that is fully elastic. Taking advantage of the tunable bending and stretching deformation enabled by the geometry of the building blocks and the semi-soft elasticity of the nematic LCE in the metamaterials, we can engineer the local stretch and stress at an unmet level of their counterpart composed by elastomers. Numerical simulations and analytical models are developed to relate the metamaterial geometries and the LCE soft elasticity to the mechanical responses. In addition, an elastic region with near-zero stiffness up to a stretch of 1.4 can be designed by connecting the compliant responses due to bending deformation and the soft elasticity in the LCE. We expect that the specialized mechanical tunability enabled by the LCE metamaterials can facilitate the development of advanced forms of mechanical metamaterials and impact the design of robotic systems.

Probing the in-plane liquid-like behavior of liquid crystal elastomers

When isotropic solids are unequally stretched in two orthogonal directions, the true stress (force per actual cross-sectional area) in the larger strain direction is typically higher than that in the smaller one. We show that thiol-acrylate liquid crystal elastomers with polydomain texture exhibit an unusual tendency: The true stresses in the two directions are always identical and governed only by the area change in the loading plane, independently of the combination of imposed strains in the two directions. This feature proves a previously unidentified state of matter that can vary its shape freely with no extra mechanical energy like liquids when deformed in the plane. The theory and simulation that explain the unique behavior are also provided. The in-plane liquid-like behavior opens doors for manifold applications, including wrinkle-free membranes and adaptable materials.

Effect of stretching angle on the stress plateau behavior of main-chain liquid crystal elastomers

The equilibrium nonlinear stress-stretch relationships for a monodomain main-chain nematic elastomer (MNE) are investigated by varying the angle between the stretching and initial director axes (θ₀). Angle θ₀ has pronounced effects on the ultimate elongation as well as on the width of the low stress plateau regime (Λ_p) during director rotation, whereas θ₀ has no appreciable effect on the plateau stress (σ_p). In the stretching normal to the initial director (θ₀ = 90°), the plateau end exceeds 200% strain. At oblique angles of 90° > θ₀≥ 35°, Λ_p decreases with decreasing θ₀, whereas the definite plateau regime vanishes at θ₀ < 24°. Wide-angle X-ray scattering and polarized optical microscopy measurements reveal that the director rotates uniformly in the biased direction for the MNE of θ₀°≪ 90°, whereas directors rotating clockwise and counterclockwise are coexistent for θ₀ = 90°. Over the entire plateau regime, the MNEs exhibit pure shear deformation characterized by a Poisson's ratio of zero in the direction of the rotation axis. The Λ_p for the corresponding polydomain NE (PNE) undergoing a transition to the monodomain alignment is smaller than that of the MNE of θ₀ = 90°, while the σ_p values for both NEs are almost similar. The semi-soft elasticity concept satisfactorily explains the effects of θ₀ on Λ_p, and the Λ_p value of the PNE, using a single anisotropy parameter which is evaluated from the degree of thermally induced deformation of MNEs.

A blend of stretching and bending in nematic polymer networks

Nematic polymer networks are (heat and light) activable materials, which combine the features of rubber and nematic liquid crystals. When only the stretching energy of a thin sheet of nematic polymer network is minimized, the intrinsic (Gaussian) curvature of the shape it takes upon (thermal or optical) actuation is determined. This, unfortunately, produces a multitude of possible shapes, for which we need a selection criterion, which may only be provided by a correcting bending energy depending on the extrinsic curvatures of the deformed shape. The literature has so far offered approximate corrections depending on the mean curvature. In this paper, we derive the appropriate bending energy for a sheet of nematic polymer network from the celebrated neo-classical energy of nematic elastomers in three space dimensions. This task is performed via a dimension reduction based on a modified Kirchhoff-Love hypothesis, which withstands the criticism of more sophisticated analytical tools. The result is a surface elastic free-energy density where stretching and bending are blended together; they may or may not be length-separated, and should be minimized together. The extrinsic curvatures of the deformed shape not only feature in the bending energy through the mean curvature, but also through the relative orientation of the nematic director in the frame of the directions of principal curvatures.

Tunable wrinkling of thin nematic liquid crystal elastomer sheets

Instabilities in thin elastic sheets, such as wrinkles, are of broad interest both from a fundamental viewpoint and also because of their potential for engineering applications. Nematic liquid crystal elastomers offer a new form of control of these instabilities through direct coupling between microscopic degrees of freedom, resulting from orientational ordering of rodlike molecules, and macroscopic strain. By a standard method of dimensional reduction, we construct a plate theory for thin sheets of nematic elastomer. We then apply this theory to the study of the formation of wrinkles due to compression of a thin sheet of nematic liquid crystal elastomer atop an elastic or fluid substrate. We find the scaling of the wrinkle wavelength in terms of material parameters and the applied compression. The wavelength of the wrinkles is found to be nonmonotonic in the compressive strain due to the presence of the nematic. Finally, due to soft modes, the critical stress for the appearance of wrinkles can be much higher than in an isotropic elastomer and depends nontrivially on the manner in which the elastomer was prepared.

Main-chain liquid-crystal elastomers versus side-chain liquid-crystal elastomers: similarities and differences in their mechanical properties

After a general introduction on the main aspects of the mechanical properties of main-chain liquid-crystal elastomers (MCLCEs) and side-chain liquid-crystal elastomers (SCLCEs), new results will be presented dealing with several MCLCEs with a cross-linker density C = 8%, 6% and 4% and with a SCLCE with C = 10%, all prepared by the two-step cross-linking process. A non-SCLCE with bulky side-groups similar in shape to the mesogens was also synthesized for comparison with the SCLCE. Most of the experiments were performed with a piezorheometer allowing the determination of the shear anisotropy of the samples by applying shear in a direction parallel or perpendicular to the director, and with a thermo-elastic device for the E measurements. The main results concern: (a) the influence of the supercritical nature of SCLCE and the subcritical nature of MCLCEs on the mechanical properties of these elastomers, as well as that of SmC domains present in MCLCEs; (b) the relationship between the degrees of elongation and of anisotropy deduced from the variations of and during the poly-domain to mono-domain transition of the 10% SCLCE and the 8% MCLCE; (c) the determination of the Poisson's ratio showing that it is isotropic for the non-SCLCE, with a crossover between 0.5 (classical value for rubbers) for small strains and 0.38 for high strains, and anisotropic for the 10% SCLCE and 8% MCLCE, with values <0.5. The particular behaviors of the Poisson's ratios can be explained by confinement effects occurring when stretching increases.

Synthesis of Elastomeric Liquid Crystalline Polymer Networks via Chain Transfer

Materials capable of complex shape changes have broad reaching applications spanning biomimetic devices, componentless actuators, artificial muscles, and haptic displays. Liquid crystal elastomers (LCE) are a class of shape programmable materials which display anisotropic mechanical deformations in response external stimuli. This work details a synthetic strategy to quickly and efficiently prepare LCEs through the usage of chain transfer agents (CTA). The polyacrylate materials described herein exhibit large, reversible shape changes with strains greater 475%, rivalling properties observed in polysiloxane-based networks. The approach reported here is distinguished in that the materials chemistry is readily amenable to surface alignment techniques. The facile nature of the materials chemistry and the compatibility of these materials with directed self-assembly methods could further enable paradigm shifting end uses as designer substrates for flexible electronics or as actuating surfaces.

Theory of liquid crystal elastomers and polymer networks : Connection between neoclassical theory and differential geometry

In liquid crystal elastomers and polymer networks, the orientational order of liquid crystals is coupled with elastic distortions of crosslinked polymers. Previous theoretical research has described these materials through two different approaches: a neoclassical theory based on the liquid crystal director and the deformation gradient tensor, and a geometric elasticity theory based on the difference between the actual metric tensor and a reference metric. Here, we connect those two approaches using a formalism based on differential geometry. Through this connection, we determine how both the director and the geometry respond to a change of temperature.

Nematic fluctuations and semisoft elasticity in swollen liquid-crystal elastomers

Dynamic light scattering (DLS) experiments were performed on stretched sheets of liquid crystal elastomers (LCEs) swollen with a nematic solvent with different swelling ratios. We show that the obtained stress-strain curve and DLS data can still be explained with the concepts of semisoft elasticity. The stress-strain curve shows a typical semisoft response with a threshold strain and a plateau region where stress increases only a little with the applied strain. The width of the plateau decreases with the increase of the swelling ratio because the polymer backbone anisotropy reduces during the swelling. The relaxation rate of thermally excited director fluctuations, however, still shows a typical response, and our measurements indicate the presence of a soft dynamic director-shear mode, as predicted by the theory of semisoft elasticity.

On the nonlinear behaviour of nematic single crystal elastomers under biaxial mechanic and electrical force fields

A slab of nematic-side-chain-liquid-single-crystal elastomer (NSCLSCE), with the director initially oriented in the z -direction, is subjected to a pair of equal mechanical loads and electrical force fields in the x , y directions. The electric fields tend to make easier the rotation of the director after the application of the mechanical force field. A nonlinear expression for the free energy density is used to obtain the interval of stretching for which the system becomes unstable. However, the elastic energy of the network is assumed to be linear. The stress-strain curves predicted by the model show an unstable zone between two linearly increasing segments. The possibility of bifurcation phenomena has been examined.

Nematic fluctuations and semisoft elasticity in liquid-crystal elastomers

We give a detailed theory of nematic fluctuations in liquid-crystal elastomers (LCEs) and calculate relaxation rates as obtained by dynamic light scattering (DLS). In ideal LCEs, a nematic state is formed by a spontaneous orientational symmetry breaking of an isotropic state, manifesting itself in an existence of a coupled director-shear soft mode (Goldstone mode). The relaxation rate of the soft mode (a pure bend and a pure splay mode) goes to zero in a long-wavelength limit. In a real, nonideal sample with a locked-in anisotropy, on the other hand, the relaxation rates of these modes become finite. Nonideal elastomers are characterized by a plateau in the stress-strain curve, and the soft mode can be detected only upon stretching to the point of elastic instability at which the director starts to rotate. We use the semisoft model of Gaussian elasticity to derive relaxation rates as a function of deformation for different scattering geometries. We show that the bend-mode relaxation rate goes to zero at the threshold strain, so it is the soft mode. The splay mode, on the other hand, is not soft because the relaxation rate is finite at the threshold strain. We provide experimental evidence and compare DLS measurements of splay and bend modes of two side-chain LCE samples differing in crosslinking densities. Results of both samples are in complete agreement with the predictions of the semisoft model, which indicates that director relaxation properties are not influenced much by the crosslinking conditions.

Negative Poisson's ratio and semisoft elasticity of smectic-C liquid-crystal elastomers

Models of smectic-C liquid-crystal elastomers predict that they can display soft elasticity, in which the shape of the elastomer changes at no energy cost. The amplitude of the soft mode and the accompanying shears are dependent on the orientation of the layer normal and the director with respect to the stretch axis. We demonstrate that in some geometries the director is forced to rotate perpendicular to the stretch axis, causing lateral expansion of the sample-a negative Poisson's ratio. Current models do not include the effect of imperfections that must be present in the physical sample. We investigate the effect of a simple model of these imperfections on the soft modes in monodomain smectic-C elastomers in a variety of geometries. When stretching parallel to the layer normal (with imposed strain) the elastomer has a negative stiffness once the director starts to rotate. We show that this is a result of the negative Poisson's ratio in this geometry through a simple scalar model.

Mechanical properties of monodomain nematic side-chain liquid-crystalline elastomers with homeotropic and in-plane orientation of the director

We present the first study of the shear mechanical properties of monodomain nematic side-chain liquid-crystal elastomers (SCLCEs) prepared by cross-linking with UV irradiation a nematic side-chain liquid-crystal polymer oriented with an electric or a magnetic field. Their elastic behavior was studied in the dry, swollen and stretched states, in order to check the various theoretical descriptions of these systems. The shear measurements taken on the dry samples show that the shear anisotropy is much smaller than that of the usual twice cross-linked samples oriented by a mechanical stretching of the network formed after the first cross-linking step, demonstrating that the elasticity of the networks strongly depends on the preparation procedure used. The shear experiments performed on the swollen state of these two different types of elastomers reveal that the elasticity of the network is Gaussian for the elastomers oriented with the electric or the magnetic field, and non-Gaussian for the elastomers oriented with the usual stretching procedure. The analysis of the stress-strain curves of both types of elastomers with the neoclassical model based on Gaussian rubber elasticity confirms the Gaussian and non-Gaussian nature of their elasticity. The shear experiments performed as a function of the elongation of the homeotropically oriented elastomer when the shear is applied in a direction parallel to the elongation, do not show the decrease of the associated shear modulus, which is theoretically expected when the strain approaches the threshold value marking the beginning of the elastic plateau. However, the observation of this effect could be prevented by possible small misalignments of the director, as suggested by a calculation presented in one of the theories describing this effect.

Strain analysis of a chiral smectic-A elastomer

We present a detailed analysis of the molecular packing of a strained liquid crystal elastomer composed of chiral mesogens in the smectic-A phase. X-ray diffraction patterns of the elastomer collected over a range of orientations with respect to the x-ray beam were used to reconstruct the three-dimensional scattering intensity as a function of tensile strain. We show that the smectic domain order is preserved in these strained elastomers. Changes in the intensity within a given scattering plane are due to reorientation, and not loss, of the molecular order in directions orthogonal to the applied strain. Incorporating the physical parameters of the elastomer, a nonlinear elastic model is presented to describe the rotation of the smectic-layered domains under strain, thus providing a fundamental analysis to the mechanical response of these unique materials.

Elastic response and Ward identities in stressed nematic elastomers

Nematic elastomers exhibit a rich elastic response to external stresses. Of particular interest is the semisoft response of elastomers with an anisotropy direction (z) frozen in by a double cross-linking process. This response is characterized by a stress-strain curve for stresses along x perpendicular to z that rises initially, exhibits a nearly flat plateau between two critical values of strain, and then rises again. This paper explores elastic response in semisoft elastomers as a function of externally applied strain. It derives general Ward identities for elastic moduli and shows that the elastic modulus measuring response to xz shears vanishes at the boundaries of the semisoft plateau whereas moduli measuring response to shears perpendicular to the xz plane do not. It then calculates all relevant moduli in a simple model of elastomers and verifies the general Ward-identity predictions.

Strain dependence of the nematic fluctuation relaxation in liquid-crystal elastomers

Dynamic light scattering on a nematic liquid-crystal elastomer was performed as a function of deformation perpendicular to the director and along the director. We show that the relaxation rate of the nematic director fluctuations increases with strain along the director, as expected from the theory of semisoft elasticity. Deformation applied perpendicular to the director, on the other hand, decreases the relaxation rate to a very small value at the onset of the soft elastic response, revealing the existence of a dynamic soft mode. The results are in complete agreement with the theory of semisoft elasticity and allow us to determine all the constants of the model.

Response of prestretched nematic elastomers to external fields

We investigate the response of prestretched nematic side-chain liquid single-crystal elastomers to superimposed external shear, electric, and magnetic fields of small amplitude. The prestretching direction is oriented perpendicular to the initial nematic director orientation, which enforces director reorientation. Furthermore, the shear plane contains the direction of prestretch. In this case, we obtain a strongly decreased effective shear modulus in the vicinity of the onset and the completion of the enforced director rotation. For the same regions, we find that it becomes comparatively easy to reorient the director by external electric and magnetic fields. These results were derived using conventional elasticity theory and its coupling to relative director-network rotations.

Observation of a soft mode of elastic instability in liquid crystal elastomers

In monodomain liquid crystal elastomers a symmetry-breaking locked-in anisotropy causes a semisoft elastic response characterized by a plateau in the stress-strain curve. We show by dynamic light scattering performed as a function of deformation that the relaxation rate of the nematic director fluctuations decreases with strain to a very small value at the onset of the soft elastic response, revealing the existence of a dynamic soft mode. The results are in complete agreement with the theory of semisoft elasticity and allow us to determine all the constants of the model.

Supersoft elasticity in polydomain nematic elastomers

We consider the equilibrium stress-strain behavior of polydomain liquid crystal elastomers (PLCEs). We show that there is a fundamental difference between PLCEs cross-linked in the high temperature isotropic and low temperature aligned states. PLCEs cross-linked in the isotropic state then cooled to an aligned state will exhibit extremely soft elasticity (confirmed by recent experiments) and ordered director patterns characteristic of textured deformations. PLCEs cross-linked in the aligned state will be mechanically much harder and characterized by disclination textures.

Elastic energies for nematic elastomers

We discuss several elastic energies for nematic elastomers and their small strain expansions both in the regime of large director rotations, and in the case that director changes are small. We propose two fully non-linear model anisotropic energies and compare the behavior they predict with the currently available experimental evidence.

Strain-induced polarization in non-ideal chiral nematic elastomers

Symmetry arguments are advanced that, although ideal chiral nematic elastomers cannot show strain-induced electrical polarization, non-ideal ones can. Phenomenological arguments are then presented, which predict a simple and universal form for the direction and strain dependence of the polarization. A microscopic minimal model is also developed, which predicts the same form. Finally, an example of a polarization–strain curve is calculated for a typical experimental geometry. In this geometry, the polarization is exactly zero at both small and large strains, but pronounced for a large set of intermediate strains corresponding to the strains that cause incremental rotation of the nematic director.